In this paper we consider the evolution of boundaries of sets by a
fractional mean curvature flow. We show that, for any dimension n ≥ 2, there
exist embedded hypersurfaces in Rn which develop a singularity without
shrinking to a point. Such examples are well known for the classical mean
curvature flow for n ≥ 3. Interestingly, when n = 2, our result provides
instead a counterexample in the nonlocal framework to the well known
Grayson's Theorem [17], which states that any smooth embedded curve in the
plane evolving by (classical) MCF shrinks to a point. The essential step in
our construction is an estimate which ensures that a suitably small
perturbation of a thin strip has positive fractional curvature at every
boundary point.
